Fractions
Suggested Time: 3 Weeks
199
FRACTIONS
Unit Overview
200
Focus and Context
In Grade 6, students will deepen their understanding of number by
continuing to work on fractions. They will develop an understanding of
fractions that are greater than one and how these improper fractions can
be expressed as mixed numbers. Students will have many opportunities
to develop these concepts as they model, draw, name and write about
mixed numbers and improper fractions while solving meaningful
problems. These ‘hands on’ experiences will enable students to visualize
the connections between improper fractions and mixed numbers.
The study of improper fractions and mixed numbers should build on
students’ prior knowledge of whole number and proportional concepts
and skills and their encounters with decimals, proper fractions, ratios
and percents in previous work and in their everyday life. Students
will be engaged in problem solving situations where they will have to
compare improper fractions and mixed numbers. As they work through
various situations, students will continue to develop their problem
solving strategies as they learn how to effectively use models, pictures
and logical reasoning to solve problems.
Math Connects
Fractions are a part of everyday life; cooking, measuring, building
and understanding quantities are just a few examples of where people
encounter fractions. Learning about fractions that have a value greater
than one becomes merely an extension of what students have already
learned previously. This not only helps in the development of a strong
sense of number, but also gives students a better understanding of
how these numbers are used in everyday life. Thus, developing this
number sense, understanding what a number means, what it represents
and how it can be used becomes a reason to understand and become
familiar with the concepts of improper fractions and mixed numbers.
As students learn about these concepts through solving problems in
context, they can consider the advantages and disadvantages of various
representations of these quantities. For example, they could consider
the reasonableness of showing how much flour is needed to make pizza
dough as a mixed number rather than an improper fraction. Their
use of logical reasoning to help solve problems such as this becomes
stronger as they add more strategies to helping them solve everyday, real
world problems.
grade 6 mathematics Curriculum Guide - INTERIM
FRACTIONS
Process Standards
Key
Curriculum
Outcomes
[C]
[CN]
[ME]
Communication
[PS] Problem Solving
Connections
[R] Reasoning
Mental Mathematics [T] Technology
and Estimation [V] Visualization
STRAND
OUTCOME
Number
6N4 Relate improper
fractions to mixed
numbers.
grade 6 mathematics Curriculum Guide - INTerim
PROCESS
STANDARDS
[CN, ME, R, V]
201
FRACTIONS
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
6N4. Relate improper fractions
to mixed numbers.
[CN, ME, R, V]
Achievement Indicator:
6N4.1 Demonstrate, using
models, that a given improper
fraction represents a number
greater than 1.
Improper fractions and mixed numbers are new to Grade 6 students.
This work provides students with opportunities to build on their prior
understanding of equivalent fractions and their ability to compare
fractions with like and unlike denominators where the numerator is less
than the denominator. In Grade 6, students now work on fractions that
are greater than 1 and relate this to mixed numbers. This is completed
through the use of models and pictures such as pattern blocks, fraction
pieces and number lines.
Students will have an understanding of fractional parts, or equal shares
which they refer to as thirds, fourths, fifths, tenths, etc. They will
recognize that these fractional parts can be counted in the same way
as any other set of objects. Fractions greater than one whole can be
understood this way. For example, six fourths is just six parts called
fourths. By counting fractional parts we can help students develop
a completely generalized system for naming fractions before they
learn about fraction symbolism. Counting fractional parts lays the
groundwork for several important ideas. The idea that fifths are smaller
than halves, for example, can be a difficult concept for students to
understand since 5 is usually ‘more’ than 2. Counting the different
size parts and seeing how many parts it takes to make one whole is an
opportunity to begin thinking on this idea.
To illustrate this idea, show students manipulatives representing five or
six fourths. Ask students to tell how many fourths. Ask if the collection
is more or less than one whole, two wholes? While doing this, prompt
students to make informal comparisons among different counts. For
example, ask them to explore the reasons they would get almost two
wholes out of eight fifths when they would not get one whole out of five
eighths. Ask them questions such as “What is another way we could say
seven fourths? To introduce this topic, use fractional parts that students
are most comfortable with, such as thirds, fourths and halves. Challenge
students to come up with ways to represent what the fractional pieces
5
show. For example, if they have 4 , ask students to explore other ways
of showing how they are thinking about this fraction using models,
pictures and numbers.
Models can help students clarify ideas that are often confused in a purely
symbolic mode. Sometimes it is useful to do the same activity with two
different models. From the view point of the student, the activity may
be quite different. Using two different models, for example pattern
blocks and fraction tiles, students visually see how the same fraction can
be modelled in two different ways. This may help students understand
the concept of improper fractions holding a value that is greater than 1.
(continued)
202
Grade 6 mathematics Curriculum Guide - INterim
FRACTIONS
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Performance
Math Focus 6
• Tell students Billy went to the store. While he was there he noticed
a sign saying bars were on sale. “4 for $1.00. Billy thought this was
a great deal and bought 4 bars. Billy wanted to share the bars among
his two friends. Show how Billy can share this equally among him
and his two friends.
(6N4.1, 6N4.2)
Lesson 1: Modelling Fractions
6N4
TG pp. 13 – 17
Student-Teacher Dialogue
• Ask students to explain why
5
3
must be greater than one whole.
(6N4.1, 6N4.2)
• Ask students to model
7
4
to show that it is greater than one whole.
Ask students to model an improper fraction. Ask them to explain
how they know it is an improper fraction.
(6N4.1)
9
• Tell students you modelled the improper fraction 6 using pattern
blocks. Ask them how they know you used 9 blocks. Ask them to
9
determine which pattern block was used to model 6 and explain how
they know.
(6N4.1)
10
• Tell students that the improper fraction 3 can be modelled using
pattern blocks. How many different ways can you model this
improper fraction using these pattern blocks?
(6N4.1)
• Ask students to represent the following improper fractions using
rectangles:
(i)
(ii)
5
4
3
2
Ask students to explain their thinking.
grade 6 mathematics Curriculum Guide - INTERIM
(6N4.1)
203
FRACTIONS
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
6N4 Continued
Achievement Indicator:
6N4.1 Continued
Making meaningful connections to the world can also help students
understand the concepts they are learning. While talking about
improper fractions, and even mixed numbers, money can be used to
help students understand these concepts. A dime, for example can
be seen as 1 of a dollar where 10 dimes make one whole dollar. Use
10
this context to ask students to explore ways to make more than one
dollar using these coins. For example, if they use 5 quarters, which
5
can be represented as 4 , they will have $1.25 or 1 1 of a dollar. Since
4
students are really familiar with money, it may be a good place to start
when talking about these concepts. This can also help students see the
connections between improper fractions and mixed numbers.
It is important that students have a strong conceptual understanding of
improper fractions. Students need to be able to understand and explain
that an improper fraction represents more than one whole and that its
numerator is greater than its denominator. To create this conceptual
understanding it is suggested that students engage in many hands
on activities that require them to solving problems using improper
fractions. Using this approach and encouraging the use of models and
pictures, students will begin to get a good idea about what improper
fractions and mixed numbers really mean.
Two central concepts students need to develop about improper fractions
is that of the purpose of the denominator and the numerator and the
fact that the number of parts in a whole may change depending on the
situation. Remind students that the denominator names the number
of parts in a whole and the numerator names how many parts in all.
Similarly, students also need to recognize that the number of parts in a
whole can change. Doing activities such as using pattern blocks with the
yellow hexagon as being one whole and then changing the whole to be
two yellow hexagon, can help students understand this concept.
1
When using one hexagon as a whole, the red trapezoid is 2 , the
1
blue rhombus is 1 and the green triangle is 6 .
3
When two yellow hexagons becomes one whole, the red trapezoid,
1
1
then becomes 4 , the blue rhombus becomes 6 , and the green
1
triangle becomes 12 .
It may be a good idea to do several examples of this using these blocks to
help students recognize this difference.
204
Grade 6 mathematics Curriculum Guide - INterim
FRACTIONS
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Journal
Math Focus 6
• Write the following on the board “All improper fractions must be
greater than one whole”. Ask students to explore this statement using
models and pictures to agree or disagree with the statement. Ask
them to justify or prove their thinking about the statement.
Lesson 2: Fractions Greater Than
1
(6N4.1, 6N4.2)
6N4
TG pp. 18 - 22
Performance
• Tell students there are 12 eggs in a carton. Ask them if they had 17
eggs, would 1 1 cartons be enough to hold the eggs? (6N4.1, 6N4.2)
2
• Ask students to use any type of manipulative available to model an
improper fraction. Ask students to pass this model to their shoulder
partner where their partner will determine if the model represents
an improper fraction. Ask each partner to explain how they know it
is or is not an improper fraction and use a picture and the symbolic
form to explain.
(6N4.1, 6N4.2)
• Using snap cubes, show students a model of a whole. (E.g., 5 same
coloured snap cubes would equal one whole) Ask students to explore
different ways to create an improper fraction using this whole that
would come between 1 and 2.
(6N4.1)
• Using the numbers 2, 5, 7, 8, ask students to create as many
improper fractions as they can. Ask them to choose one improper
fraction and represent it using a model, picture and in symbolic
form.
(6N4.1, 6N4.2)
grade 6 mathematics Curriculum Guide - INTERIM
205
FRACTIONS
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
6N4 Continued
Achievement Indicator:
6N4.2 Translate a given improper
fraction between concrete,
pictorial and symbolic forms.
To help students understand that an improper fraction represents a
number greater than one, use a hundreds grid. Presenting the idea that
100 blocks in the grid equals one whole (grid), students can explore
ways they can represent more than one whole. In the example below,
students are asked to shade in 245 blocks and use their picture to name
the improper fraction. Using this type of activity will also help students
strengthen their understanding of the purpose of the denominator and
numerator.
Once students have had ample opportunity building, creating,
modeling, drawing, and naming improper fractions, they will be ready
to use symbolic forms to represent the fractions they are working with.
Students should be able to easily translate a given improper fraction
between various representations such as models, pictures and then in
numbers. Ask students to model an improper fraction and then draw
this representation. Ask students to then use numbers to name this
fraction.
To help students see the relevance of translating their models of
improper fractions to pictures and then to a symbolic form ask students
1
8
to prove that 6 is less than 1 2 . To do this ask students to use pattern
8
blocks to build 6 and then draw this on paper as a part of their written
response to the question. Students could then go on to show how their
1
8
picture of the pattern blocks show that 6 is less than 1 2 .
206
Grade 6 mathematics Curriculum Guide - INterim
FRACTIONS
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Performance
• Ask students to explore the different possibilities of creating an
improper fraction with two hexagons being one whole. Challenge
them by limiting the number of blocks they can use. For example,
if two hexagons equals one whole, ask students to find various ways
to represent an improper fraction with 7 blocks. Ask them to record
their answers in a chart. Make this a class challenge where students
can find as many representations as possible, receiving one point for
each representation found and 5 points for each representation that is
unique among the students. The winner would be the student with the
most points.
(6N4.2)
Resources/Notes
Math Focus 6
Lesson 2 (Cont’d): Fractions
Greater Than 1
6N4
TG pp. 18 - 22
• Provide index cards and ask students to create cards that name an
improper fraction and that represent the improper fractions. E.g.,
5
students can draw five rhombuses to represent 3 when one hexagon
is a whole and then create the corresponding card with the symbolic
5
form of 3 . Students can then combine their completed cards and play
a matching game whereby they have to match the picture with the
number.
(6N4.2)
grade 6 mathematics Curriculum Guide - INTERIM
207
FRACTIONS
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
6N4 Continued
Achievement Indicator:
6N4.3 Express improper fractions
as mixed numbers.
After working with improper fractions, students are formally introduced
to mixed numbers. Students need to see the connection between
improper fractions and mixed numbers as both represent numbers
greater than one whole. They also need to recognize that every improper
fraction can be converted to a mixed number and every mixed number
can be converted to an improper fraction.
Pattern blocks can be used to help students see these connections.
7
Ask students to model 2 when one yellow hexagon equals one whole.
As they build this model they will see that they have created 3 whole
hexagons with one piece or one half of another hexagon left over. Ask
students to then think about how they could report what they see. Ask
them to think of another way to represent this improper fraction.
Using effective math language while modeling these concepts may
become important with regards to student understanding. E.g., if a
7
student was asked to model seven over two, or 2 , he/she may not have
a good understanding of what this really means. Compare this to asking
a student to model seven halves. As long as a student has a good sense
of what a half is, it may be easier to think about seven halves rather than
seven over two.
Through working with these numbers, students may discover the
relationship in multiplying the denominator with the whole number
and adding the numerator to get the improper fraction, but it is not the
recommended way to introduce or teach the topic.
“There is absolutely no reason ever to provide a rule about multiplying
the whole number by the bottom number and adding the top number.
Nor should students need a rule about dividing the bottom number into
the top to convert fractions to mixed numbers.” Van de Walle (2006)
p. 141.
Providing students with ample opportunities to explore these concepts
through the use of hands on activities, using models and pictures will
help students develop an understanding but in their own words and in
their own way.
(continued)
208
Grade 6 mathematics Curriculum Guide - INterim
FRACTIONS
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Math Focus 6
Performance
7
4
• Tell students: Susan wrote the improper fraction . Ask students to
think about what this improper fraction could represent.
(6N4.1, 6N4.3, 6N4.2)
9
• Ask students to think of a mixed number that is a little less than 4 .
Ask them to show how they know.
(6N4.1, 6N4.3, 6N4.6)
Lesson 3: Representing Improper
Fractions as Mixed Numbers
6N4
TG pp. 23 - 27
15
• Ask students to make a model to represent 6 . Ask them to explain
how this number can be expressed as a mixed number using models,
picture and numbers.
(6N4.3, 6N4.4)
• Ask students the following: If Daniel ate 9 half donuts, how do you
know he ate between 4 and 5 full donuts? Use models, pictures and
numbers to show your thinking.
(6N4.3, 6N4.2 6N4.4)
Additional Reading (provided):
Van de Walle, John and Lovin,
LouAnn H. (2006) Teaching
Student–Centered Mathematics.
Grades 3-5 pp. 131-144.
grade 6 mathematics Curriculum Guide - INTERIM
209
FRACTIONS
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
6N4 Continued
Achievement Indicators:
6N4.3 Continued
2
Give students a mixed number such as 3 5 . Their task is to find a
single fraction that names the same amount. They may use any
familiar materials or make drawings, but they must be able to give
an explanation for their result. Similarly, ask students to start with a
17
fraction greater than 1, such as 4 , and ask them to determine the mixed
number and provide a justification for their result. This activity can be
repeated using other numbers. Similarly, students may find it easier if
they are able to choose the numbers they work with for this task. (Van
de Walle, 2006).
Give students 5 green triangles and using the yellow hexagon as a whole.
Ask students to write the mixed number that represents the triangles.
(The purpose of this activity is to help students realize that in order
to write a mixed number, the numerator must be greater than the
denominator, or there has to be at least one full whole in the number.)
6N4.4 Translate a given mixed
number between concrete,
pictorial and symbolic forms.
Provide students with opportunities to use concrete, pictorial and
symbolic forms to represent numbers. This helps students become
exposed to the number in more than one way, where they are physically
working with the number using materials. Drawing a picture to
represent the number they are working with helps students solidify the
concrete image of the number. The next step in this progression of
learning is naming the number using symbols.
2
Ask students to model a given mixed number, for example 3 and 6
using manipulatives. You may wish to give them a choice of several
different mixed numbers depending upon their understanding of mixed
numbers. Allow students time to discuss their choice of manipulatives
and how their model represents their chosen mixed number. Ask
students to then draw a picture to represent this number (it could be
a picture of the model they already used, or the number in a different
context). Again, ask them to explain how their drawings represent the
given mixed number. Ask them to then represent this mixed number in
symbolic form (using numbers).
(continued)
210
Grade 6 mathematics Curriculum Guide - INterim
FRACTIONS
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Math Focus 6
Performance
4
3
• Tell students that a cookie recipe calls for cups of flour. Mr.
Bob is not sure what this means. Ask students to help Mr. Bob by
4
explaining to him what 3 means and tell him how much flour he
needs using a mixed number.
(6N4.3)
• Ask students to model and then draw a picture to show that
5
2
=
2
1
.
2
Lesson 3 (Cont’d): Representing
Improper Fractions as Mixed
Numbers
6N4
TG pp. 23 - 27
(6N4.3, 6N4.5, 6N4.2, 6N4.4)
• Tell students to explain a situation when it would be a good idea to
express an improper fraction as a mixed number.
(6N4.3)
grade 6 mathematics Curriculum Guide - INTERIM
211
FRACTIONS
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
6N4 Continued
Achievement Indicators:
6N4.3 Continued
6N4.4 Continued
As students continue to work with improper fractions and mixed
numbers, they may begin to use and refine personal strategies they
have created to make sense of this concept. Encourage the use of these
strategies and provide students with a lot of opportunities to talk about
the strategies they are using and ask them to demonstrate these to other
students.
Many types of manipulatives can be used to explore improper fractions
and mixed numbers. For example, coloured snap cubes can be used to
create different lengths where students can compare the lengths of the
joined cubes. Students could create a length of 4 blue cubes and another
with 8 yellow cubes. Students could then talk about how the lengths
1
of these two compare. So, blue is 2 the length of yellow. If they then
2
create a length of 6 red cubes, they would say that red is 1 4 the length
of blue.
Another suggested activity in using the snap cubes would be to provide
students with 20 same colored snap cubes. Ask students to represent an
17
improper fraction such as 5 . They should understand from previous
17
work that 5 means there are 5 in a whole with 17 parts in all. They
should then go on to create towers of 5 snap cubes where they will see
that they can create 3 complete towers with 2 cubes left over. This can
17
2
be used then to help them see that 5 is the same as 3 and 5 .
212
Grade 6 mathematics Curriculum Guide - INterim
FRACTIONS
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Performance
Math Focus 6
• Invite students to make a plan to teach their parents about improper
fractions and mixed numbers. Ask students to use models, pictures,
numbers and words to show their parents how to express a mixed
number as an improper fraction.
(6N4.3, 6N4.5)
Lesson 4: Exploring Improper
Fractions and Mixed Numbers
6N4
TG pp. 28 – 31
• Improper Fraction War - Using a deck of cards containing numerals
1-9, ask students to work in small groups. Shuffle the cards and deal
4 cards to each player. Students can use any two of the 4 cards in
their hand to create the greatest possible improper fraction. In turn,
each player will reveal their improper fraction determining who has
the greatest number. Students may have to convert these improper
fractions to mixed numbers to help them compare the numbers. The
player with the greatest improper fraction will score one point. The
first player with 5 points wins.
(6N4.3)
• Ask students to create their own riddles about improper fractions
and mixed numbers. Ask their classmates to solve the riddles. E.g.,
I am a unit of time.
The lesson activity calls for the
use of colored rods. If these are not
available, colored snap cubes can be
used instead.
Children’s Literature (provided):
Merrill, Jean The Toothpaste
Millionaire
Please note: The book is written
using Imperial units. Please modify
to reflect the metric system.
My denominator is 2.
My numerator is the number of days in one week.
How many days did Mary work?
(7/2 = 3 1/2 days)
(6N4.3, 6N4.5)
grade 6 mathematics Curriculum Guide - INTERIM
213
FRACTIONS
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
6N4 Continued
Achievement Indicator:
6N4.5 Express mixed numbers as
improper fractions.
5
Ask students to use any manipulative to model or represent 3 . Discuss
with students their choice of models and lead the discussion to help
2
5
them see or link the 3 to 1 3 . Invite students to explain some of their
5
personal strategies that help them to understand that 3 is the same as 1
2
and 3 .
Using pattern blocks, have the yellow hexagon represent 1 whole. Give
students 7 blue rhombuses and ask them how many complete hexagons
they could cover. Ask them to explore and decide what fraction of
the hexagon they could cover with the left over blue rhombus. Then
ask students to think about how to write the mixed number and the
improper fraction that the 7 blue rhombuses represent.
Students have been modeling, creating, describing, drawing and naming
improper fractions and mixed numbers. They have been exploring with
naming improper fractions as mixed numbers and mixed numbers as
improper fractions. Now, students will strengthen their understanding
that every improper fraction can be written as a mixed number and
every mixed number can be written as an improper fraction. They
will learn that when an improper fraction can be expressed as a mixed
number these numbers then are equivalent. Equivalent fractions were
taught in Grade 5.
The figure below illustrates this idea by showing that the improper
3
7
fraction 4 and the mixed number 1 4 refer to the same fraction of a
region, or area, therefore they are equivalent.
(continued)
214
Grade 6 mathematics Curriculum Guide - INterim
FRACTIONS
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Performance
Math Focus 6
• Tell students that it takes 4 – 250 mL cups to fill a 1 L bottle. If
they had 10 – 250 mL cups, how many 1 L bottles would they fill?
Represent this answer as a mixed number and an improper fraction.
Lesson 5: Representing Mixed
Numbers as Improper Fractions
(6N4.3, 6N4.5)
6N4
TG pp. 36 - 40
• Ask students to choose one mixed number and one improper
fraction. Ask them to write these numbers on separate pieces of
paper. Next ask students to create a model for each number and
draw a picture to represent each. Next, students to switch papers and
models with a partner and asks their partner to express the improper
fraction as a mixed number and express the mixed number as an
improper fraction. Finally, partners model these new numbers using
a different type of manipulative and draw their representation of the
numbers and show the number in symbolic form.
(6N4.2, 6N4.4)
• Ask students to think of possible values for
when 13/
is an
improper fraction that is between 2 and 3. Ask students to determine
if there is more than one answer and explain how they know.
(6N4.3, 6N4.5, 6N4.6)
1
• Tell students it takes 3 of an hour to bake one batch of cookies.
If they had 5 batches of cookies to bake, how long would it take.
Ask them to represent their answer using a mixed number and an
improper fraction.
(6N4.3, 6N4.5)
grade 6 mathematics Curriculum Guide - INTERIM
215
FRACTIONS
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
6N4 Continued
Achievement Indicator:
6N4.5 Continued
The figure below illustrates another way to show that an improper
fraction can be expressed as a mixed number, where both name the same
quantity. In this example, it can be seen that there is one full egg carton,
12
5
or 12 eggs and a part of another carton, namely 12 . The resulting mixed
5
number then, would be 1 12 or the equivalent improper fraction that
17
would name the amount of the set would be 12 .
Using the book Funny and Fabulous Fraction Stories ask students to read
the section on Improper Fractions and Mixed Numbers found on page
22. Set up this activity as a reader’s theatre. Students take on the roles of
the various characters but extend the scenes, whereby, the callers of the
radio station need further clarification about the nature of converting
mixed numbers to improper fractions. Students can rewrite the script
to do this. (It may not be necessary to do the second scene with Joe
and Al as this section deals with simplifying the fractions which is not
a requirement for Grade 6). Once Joe and the caller is established,
give the remaining students the role of the Math Experts in which Joe
calls upon to help out the callers “Betty” and “Pauline”. In the scenes
Betty and Pauline call Joe Trella, the Fraction Fella, to get advice on a
problem they have involving converting mixed number and improper
fractions. The Math Experts have to come up with a plan on how to
show Betty and Pauline how to convert these numbers. Encourage
students to use different ways of teaching Betty and Pauline as they
may need to see how it can be done in different ways. These scenes
should be done separately with Joe and the callers being different people
allowing all students to be involved in the lesson. You may also decide
that Joe could have a role where he is showing Betty a way to convert
and then have the others demonstrate their thinking to coincide with
Joe’s lesson. Alternatively or as a follow up, other types of situations can
be created where students role play the scenarios demonstrating their
understanding of a concept.
(continued)
216
Grade 6 mathematics Curriculum Guide - INterim
FRACTIONS
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Performance
Math Focus 6
• Mixed Number War - Using a deck of cards containing numerals
1-9, ask students to work in small groups. Shuffle the cards and deal
3 cards to each player. Students use the cards to create the greatest
possible mixed number. In turn, each player will reveal their mixed
number. The player with the greatest mixed number will score one
point. The first player with 5 points wins.
(6N4.5)
Lesson 5 (Cont’d): Representing
Mixed Numbers as Improper
Fractions
6N4
TG pp. 36 - 40
• Place a mixed number/improper fraction on students’ back. Instruct
the class to question the student about the number to determine
what it is. Limit the number of questions the student can ask.
This could also be done as a class activity where all students have
improper fractions/mixed numbers on their back where they have
to go around the class asking questions to get clues about what their
number is.
(6N4.3, 6N4.5)
• Tell students that you have 18 quarters in a coin collection. How
many dollars would this be? Can you model the amount as an
improper fraction? Mixed number? Which one would be easier to
understand how much money you have altogether? (6N4.3, 6N4.5)
• Mario and Sydney are great hockey players. To prepare for the
4
25
season, Mario practiced 3 7 weeks. Sydney practiced 7 weeks. Who
practiced more? Explain your thinking using pictures, numbers and
words.
(6N4.3, 6N4.5)
grade 6 mathematics Curriculum Guide - INTERIM
Children’s Literature
(provided):
Greenberg, Dan. Funny and
Fabulous Fraction Stories
217
FRACTIONS
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
6N4 Continued
Achievement Indicator:
6N4.5 Continued
1
Give students a mixed number such as 2 3 . Using pattern blocks, ask
students to use the blue rhombi to create a design on triangular dot
paper that represents this mixed number. Ask them to use their design to
explain how this mixed number also represents the equivalent improper
fraction, where they would write the improper fraction in symbolic
form.
This is just one example of how the blue rhombi can be used to model 2
1
. Encourage students to be creative in their designs.
3
The picture below shows another example of how to use pattern
blocks, namely the green triangles, to represent an improper fraction on
13
triangular dot paper. Ask students to represent 6 and then explain how
1
this also shows 2 6 .
218
Grade 6 mathematics Curriculum Guide - INterim
FRACTIONS
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Performance
Math Focus 6
• Ask students to choose a mixed number and tell them to keep it a
secret from their classmates. Set up centers around the room where
students could model their number in one center, draw it in another
and then represent it using an improper fraction in yet another
center. After everyone has had an opportunity to represent their
number in the three centers, bring the class together. Ask them to
match all the models to the corresponding pictures and numbers.
Lesson 5 (Cont’d): Representing
Mixed Numbers as Improper
Fractions
6N4
TG pp. 36 - 40
(6N4.2)
• Give students an equivalent improper fraction and a mixed number
2
12
(e.g., 5 and 2 5 ). Ask them to represent these two numbers
concretely, pictorially and symbolically to show they are equivalent.
(6N4.4, 6N4.5)
grade 6 mathematics Curriculum Guide - INTERIM
219
FRACTIONS
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
6N4 Continued
Achievement Indicators:
6N4.6 Place a given set of
fractions, including mixed
numbers and improper fractions,
on a number line, and explain
strategies used to determine
position.
Students have been working with mixed numbers and improper fractions
by representing, modelling, naming and expressing them in different
forms. As students continue working with these numbers they can begin
thinking about strategies that would enable them to compare improper
fractions and mixed numbers. In Grade 5, students worked with
comparing proper fractions with like and unlike denominators which will
now help them compare mixed numbers and allow them to extend their
personal strategies to compare improper fractions.
When comparing improper fractions, encourage students to recognize
that it may be easier to express the improper fraction as a mixed number
where they would compare the whole number first and then look at
9
6
the proper fraction if needed. For example, when comparing 4 and 5 ,
4
2
students could express both as a mixed number namely, 1 4 and 1 5 . Here
they could easily see that they are both 1 whole with the first having an
4
1
2
extra 4 or 2 and the second having an extra 5 . Students should be able to
9
4
1
6
see that 5 is greater than 4 because 5 is greater than 2 .
Using a number line when solving problems is another strategy
that students can use to help show their understanding. As students
encounter problems that require them to compare mixed numbers and
improper fractions, a logical representation of the understanding would
be to place the given numbers on a number line.
Fractional Clothesline - Stretch string across the classroom with various
points marked for 0, 1, 2, 3, and 4. You may want to ask students to
do this. The clothesline will be used to show students that all proper
fractions are between 0 and 1 and all mixed numbers and improper
fractions are greater than 1. Students will clip index cards with various
proper fractions, improper fractions and mixed numbers on the
clothesline. You may wish to ask students to place various benchmarks
and ask them to choose the numbers they want to place. For example,
you may wish to ask them to write any mixed number or improper
fraction that would come between 1 and 2, or 3 and 5 that could go
on the number line. After each student has had a chance to place some
numbers on the line, have a class discussion to decide if all the placed
numbers are in relative positions. You may even want to give the class
a chance to change some already placed numbers before the discussion
takes place.
(continued)
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Grade 6 mathematics Curriculum Guide - INterim
FRACTIONS
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Math Focus 6
Performance
4
3
1
3
• Ask students: Bill said he ate 1 bags of popcorn. Ed said he ate
bags of popcorn. Assuming each bag of popcorn is the same size, is it
possible for Bill to have eaten more popcorn than Ed? Explain your
thinking with pictures, numbers and words.
(6N4.3, 6N4.5, 6N4.6)
Lesson 6: Comparing Fractions
and Mixed Numbers
6N4
TG pp. 41 – 45
3
26
• Ask students to determine which number is greater – 5 or 4 4 . Ask
students to explain using pictures, numbers and words how they
know.
(6N4.6)
Math Game: Spinner Fractions
5
3
• Tell students: 4 friends were at a party. Joe stated he ate of pizza
5
while Amy stated she ate 4 of pizza. Larry said that Amy ate more
pizza than Joe. Is Larry correct? Explain your thinking.
6N4
TG pp. 46 - 47
(6N4.1, 6N4.6)
• Ask students why someone would think that
Use pictures, numbers and words to explain.
5
4
is greater than
3
2
?
(6N4.1, 6N4.3, 6N4.5, 6N4.6)
3
1
5
• On the number line below, place the following: 2 , 3 2 , 4 ,
Choose one to explain its placement.
15
,
4
2
1 5.
(6N4.6)
• Ask students how they would immediately know that 2
7
than 1 8 ? Explain.
2
5
Additional Reading (provided):
Van de Walle, John and Lovin,
LouAnn H. (2006) Teaching
Student–Centered Mathematics.
Grades 3-5 pp. 146-150.
is greater
(6N4.6)
• Ask students to choose two improper fractions or two mixed
numbers, order them and compare them. Ask them to explain to a
friend how they know they have ordered their numbers correctly.
(6N4.1, 6N4.3, 6N4.5, 6N4.6)
• Tell students two athletes were competing in a biathlon. Athlete
7
12
one finished the race in 6 hours and the other athlete finished in 10
hours. Ask students to determine who won.
(6N4.6)
grade 6 mathematics Curriculum Guide - INTERIM
221
FRACTIONS
Strand: Number
Outcomes
Elaborations—Strategies for Learning and Teaching
Students will be expected to
6N4 Continued
Achievement Indicators:
6N4.3 Continued
6N4.5 Continued
6N4.6 Continued
As students continue working with improper fractions and mixed
numbers, they can use this understanding to help them solve
meaningful problems using logical thinking. Using logical reasoning is
yet one strategy for solving problems students have developed. Engaging
students in problems that allow them to synthesize and analyze the
information presented, having them plan out their solutions and then
represent their thinking using pictures, numbers and words will help
strengthen their problem solving process.
Another strategy that may help students solve problems is that of
solving a simpler problem that is related to a given problem. For
example, if a problem deals with both improper fractions and mixed
numbers, ask students to solve the problem in which only improper
fractions are used. Similarly, students could also change the numbers in
the problem (if that is what is causing difficulty) to numbers they are
more comfortable with and note how they would go about solving the
problem using simpler numbers and how this could be transferred to
solving the problem with the given numbers.
Encourage students to pick out the most important parts of the
problem, draw or use a model to help them visualize the given
information and identify exactly what the problem is asking. Students
should be given many opportunities to work together while solving
problems and talk out their thinking with their partner, as it is through
this collaboration that students learn best.
Students may find it interesting to explore how the fraction name
changes if the whole changes. If the blue rhombus is the whole, using
1
pattern blocks, for example, the green triangle would be 2 and the red
1
3
trapezoid would be 1 2 or 2 . Playing with these different scenarios will
help students better understand the relationship between the improper
fractions and mixed numbers.
222
Grade 6 mathematics Curriculum Guide - INterim
FRACTIONS
General Outcome: Develop Number Sense
Suggested Assessment Strategies
Resources/Notes
Math Focus 6
Performance
• Tell students there are 24 cans of soft drink in one case. 60 cans of
drink are needed for the volleyball tournament. How could knowing
about mixed numbers and improper fractions help you determine
the number of cases of drinks that are needed for the tournament?
(Remind students that a part of a case of drink can be bought)
(6N4.1, 6N4.3)
Lesson 7: Solving Problems Using
Logical Reasoning
6N4
TG pp. 48 – 50
Curious Math: Growing Shapes
6N4
• Ask students to write two improper fractions and two mixed
numbers that are between 4 and 5. Ask them to explain their
thinking.
(6N4.6)
TG pp. 51 – 52
• Pose the following to students: You just baked one dozen brownies.
Seven of your friends just arrived and you want to share the brownies
equally among all of you. Show how you would do this and write the
amount you all get using a mixed number and an improper fraction.
(6N4.3, 6N4.5)
1
• Tell students that the answer to a problem is 2 3 . Ask them what the
question might be?
(6N4.6)
Student-Teacher Dialogue
• Ask students: Which of the following improper fractions and mixed
numbers are between 2 and 3? Explain.
7
3
1
1
, 2 5, 3 2,
7
4
grade 6 mathematics Curriculum Guide - INTERIM
(6N4.6)
223
FRACTIONS
224
grade 6 mathematics Curriculum Guide - INTERIM